Linear Approximations ACADEMIC RESOURCE CENTER

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1 Linear Approximations ACADEMIC RESOURCE CENTER

2 Table of Contents Linear Function Linear Function or Not Real World Uses for Linear Equations Why Do We Use Linear Equations? Estimation with Linear Approximations References

3 Table of Contents Linear Function Linear Function or Not Real World Uses for Linear Equations Why Do We Use Linear Equations? Estimation with Linear Approximations References

4 Linear Function Definition: A mathematical equation in which no independent-variable x is raised to a power greater than one. A simple linear function with only one independent variable y (y = ax + b) traces a straight line when plotted on a graph. Also known as a linear equation. Famous Forms: Y-axis form y = mx + b Point-slope form (y y 1 ) = m(x x 1 ) ( x ) ( y ) Intercept form + = 1 c b

5 Table of Contents Linear Function Linear Function or Not Real World Uses for Linear Equations Why Do We Use Linear Equations? Estimation with Linear Approximations References

6 Linear Function or Not 4y = 3x + 2 xy = 3 2x = 4y + 2 x 2 + 3y = 2 x + 3 = y 3 x + 3 = y x + y = 3x + 2 x(3 + x) = y y = 3x 3(xy + y 2 ) = 4y x 2 + y 4 = 1 4a + 3b = 6

7 Answers 4y = 3x + 2 Linear Function xy = 3 Not 2x = 4y + 2 Linear Function x 2 + 3y = 2 Not x + 3 = y 3 Not x + 3 = y Not x + y = 3x + 2 Linear Function x(3 + x) = y Not y = 3x Linear Function 3(xy + y 2 ) = 4y Linear Function x 2 + y 4 = 1 Linear Function 4a + 3b = 6 Linear Function

8 Reasoning for the Nonlinear Functions xy = 3 Not: Because the independent varialbe is multiplied to the dependent variable. x 2 + 3y = 2 Not: Because the independent variable is raised to a power other than 1. x + 3 = y 3 Not: Because the dependent variable is raised to a power other than 1. x + 3 = y Not: Because the independent variable is raised to a power other than 1. (i.e. x = x 1 2 ) x(3 + x) = y Not: Because after distribution, the indenpendent variable is raised to a power other than 1.

9 Table of Contents Linear Function Linear Function or Not Real World Uses for Linear Equations Why Do We Use Linear Equations? Estimation with Linear Approximations References

10 Real World Uses for Linear Equations Popular Uses Demand Curves (economic analysis) Interest Rates and Investments (finance industry) Foreign Currency Jobs Managers Financial Occupations Computer Programmers Scientists Engineers Administrators Construction Health Care

11 Table of Contents Linear Function Linear Function or Not Real World Uses for Linear Equations Why Do We Use Linear Equations? Estimation with Linear Approximations References

12 Why Do We Use Linear Equations? Linear Equations are used in everyday life. Calculating travel times Converting hours to minutes Weights and measures (Doubling a recipe) Estimation

13 Table of Contents Linear Function Linear Function or Not Real World Uses for Linear Equations Why Do We Use Linear Equations? Estimation with Linear Approximations References

14 Estimation with Linear Approximations Suppose we wanted to approximate 99. We could say that = 10. However, using linear approximations, we can obtain a better approximation than 10. Let us take a look at the non-linear function f (x) = x. This function represents all of the square roots. i.e. f (3) = 3. Now using Mathematica to visualize.

15 Estimation with Linear Approximations

16 Estimation with Linear Approximations Now that we have motivation, we should find a linear approximation around the point x = 100. Our reasoning is simply because we know the function value at that point and it is near 99. i.e. f (100) = 10. So we wish to find a line that passes through the function x at the point x = 100, then we will use that line to approximate the point x = 99. To start, let us take the form y = mx + b, where m is the slope and b is the y-intercept.

17 Estimation with Linear Approximations In order to determine the linear equation, we must determine what the slope of the line is. Since m = f (x), m = f (x) = 1 2 x And we wish to know the slope of a line at the point x = 100, so the slope must be f (100) = Now our equation is: y = 1 20 x + b

18 Estimation with Linear Approximations Next we must determine b. We can use the point at which we are making this linear approximation, x = 100. By plugging in 10 for y and 100 for x, we get: y = 1 20 x + b 10 = 1 20 (100) + b 10 = 5 + b 5 = b Now we have our linear approximation of f (x) = x about x = 100 in and will use it to approximate f (99). y = 1 20 x + 5

19 Estimation with Linear Approximations Using Mathematica, we can plot the function and the linear approximation together.

20 Estimation with Linear Approximations Zooming in near the point x = 100 we have:

21 Estimation with Linear Approximations Plotting the error of the two functions, we can clearly see that the linear approximation will be a good approximation for f (99).

22 Estimation with Linear Approximations We can see that the error for the linear approximation at x = 99 will be small. So then we can obtain the estimation for the 99. The actual value is given by Mathematica. This concludes the example for linear approximations. Hopefully you find more uses in everyday life for linear approximations.

23 Table of Contents Linear Function Linear Function or Not Real World Uses for Linear Equations Why Do We Use Linear Equations? Estimation with Linear Approximations References

24 References examples-equations-used-real-life.html